descriptive statistics, histograms, and normal approximations
DESCRIPTION
Descriptive Statistics, Histograms, and Normal Approximations. Math 1680. Overview. Obtaining Data Sets Descriptive Statistics Histograms The Normal Curve Standardization Normal Approximation Summary. Obtaining Data Sets. Before we can analyze a data set, we need to have a data set - PowerPoint PPT PresentationTRANSCRIPT
Descriptive Statistics, Histograms, and Normal Approximations
Math 1680
Overview
Obtaining Data Sets Descriptive Statistics Histograms The Normal Curve Standardization Normal Approximation Summary
Obtaining Data Sets
Before we can analyze a data set, we need to have a data set How far do you travel to get to class, in
miles? How tall are you?
Today, numerical data is easily stored and organized (and even analyzed) by several computer programs
Obtaining Data Sets
Notice that in its raw form, the data is difficult to deal with
By sorting the data, we can get a better picture of its distribution, or shape We are often interested in…
Where the data are centered How spread out they are With what frequency numbers appear
Obtaining Data Sets
Usually, the entire data set is too large to work with directly We want ways to summarize the data We have quantitative (numerical) and
pictorial descriptions available to us Descriptive Statistics Histograms
Descriptive Statistics
We can summarize the data set with a few simple numbers, called descriptive (or summary) statistics
The first and most often-used summary stat is the average (or mean) Represents the central tendency of the data set
Gives an idea of where the bulk of the points lie
To calculate the average, add up the values of all of the points and divide by the total number of points in the set
Descriptive Statistics
Calculate the average of the following data sets
60 60 60 60 60
18 59 60 63 100
18 35 60 87 100
605
300
5
6060606060
605
300
5
10087603518
605
300
5
10063605918
Descriptive Statistics
Despite having the same average, the three data sets are clearly different The average alone usually does not
describe data sets uniquely
Descriptive Statistics
The median is another central tendency measure The median marks the point where
exactly half of the data are less than (or equal to) the median If there are an odd number of data points,
then the median is just the number in the middle of the sorted set
Otherwise, the median is the average of the two points in the middle of the sorted set
Descriptive Statistics
Calculate the median of each data set 1 4 5 7 10 15 18
5249444438383534313131302929
2928282726242322212120202017
17171513131313101099330
5.222
2322
Descriptive Statistics
The average is like a balance point It represents the place where the data set
is equally “heavy” on both sides If there are outliers on one side of the
data set, the average will be skewed The median is more robust
What this means is that it is usually less s affected by outliers or data entry errors.
Descriptive Statistics
In a certain class of 13 students, 10 showed up the first exam, while 3 blew it off Here are the grades; in order:
Calculate the class median… Including all students Not counting those who slept in
98949387848179786855000
79
82.5
Descriptive Statistics
In a certain class of 13 students, 10 showed up the first exam, while 3 blew it off Here are the grades; in order:
Calculate the class average… Including all students Not counting those who slept in
98949387848179786855000
62.8
81.7
Descriptive Statistics
Suppose the teacher mistyped the grade of 55 as being a 15 Not counting the sleepers,
What is the new median?
What is the new average?
98949387848179786815000
98949387848179786855000
82.5
77.7
Descriptive Statistics
Earlier, we saw that the average did not necessarily uniquely describe a data set
We use the standard deviation (SD) to measure spread in a data set
When paired, the average and SD are highly effective summary statistics
Descriptive Statistics
The Root-Mean-Square (RMS) measures the typical absolute value of data points in a set Calculated by reading its name backwards
Square all entries in the data set Take their mean Take the square root of that mean
Find the average and then the RMS size of the numbers of the list 36531
Average = 0 RMS = 4
Descriptive Statistics
The SD embodies the same concept of “typical” distance Where the RMS measures typical
distance from 0, the SD measures typical distance from the data set’s average
This is accomplished by subtracting the average from every data point and then taking the RMS of the differences (or deviations from the mean)
Descriptive Statistics
1 4 5 7 10 15 has an average of 7
The deviations are then -6 -3 -2 0 3 8 Note how the subtraction process re-
centers the data set so that the average is at 0
Descriptive Statistics
Taking the RMS of the deviations gives the standard deviation Normally, about two thirds to three
quarters of a data set should be within one SD of the mean
1 4 5 7 10 15 has an average of 7 and an SD of about 4.5
Descriptive Statistics
1 4 5 7 10 15
-6 -3 -2 0 3 8
36 9 4 0 9 64
Average = 7
(1+4+5+7+10+15)/6 = 7
1-7 4-7 5-7 7-7 10-7 15-7
(36+9+4+0+9+64)/6 = 122/6 ≈ 20.3
√(20.3) ≈ 4.5
SD ≈ 4.5
(-6)2 (-3)2 (-2)2 02 32 82
Descriptive Statistics
What we had on the previous slide is called the SD of the sample. However, if the goal is to use this sample to estimate the SD of a larger population, we would divide by n-1 instead of n (where n is the number of points) and call the result Sample SD.
Most calculators actually calculate the sample SD. In general, the higher a set’s SD, the more spread
out its points are An SD of 0 indicates that every point in the data set
has the same value
Descriptive Statistics
Calculate the SD’s of the data sets 60 60 60 60 60
18 59 60 63 100
12 35 60 87 100
0
26.0
30.7
Histograms
Often, we would prefer a pictorial representation of a data set to a two-number summary
The most common way to graphically represent a data set is to draw a frequency histogram (or just histogram)
Histograms
Histograms tend to look like city skylines In a histogram, the area under the curve
between two points on the horizontal axis represents the proportion of data points between those two points
Continuing the city skyline analogy, the size of the building determines how many people live there A long, low building can house as many people as a
thin skyscraper
Histograms
To draw a histogram, we first need to organize our data into bins (or class intervals) Often, the bins are dictated to us If we get to choose them, we try to pick the bins
so that they give a fair representation of the data Then mark a horizontal axis with the bin
values, spacing them correctly
Histograms
Often, data is given in percentage form If not, divide the number of points in the bin by
the number of points in the data set to get the percentage
Draw a box for each bin so that the area of the box is the percentage of the data in that bin To get the correct height of the box, divide the
percentage of the box by the width of the bin
Histograms
Note that the average and median can be visually located on a histogram If the histogram was balanced on a see-saw, the fulcrum
would meet the histogram at the average If you draw a vertical line through the histogram so that it
splits the area in half, then the line passes through the median
On a symmetric histogram, the average and median tend to coincide
Asymmetric tails pull the average in the direction of the tail
The Normal Curve
A great many data sets have similarly-shaped histograms SAT scores Attendance at baseball games Battery life Cash flow of a bank Heights of adult males/females
The Normal Curve
These histograms are similar to one generated by a very special distribution It is called the normal distribution, and it
is identified by two parameters we are already familiar with average standard deviation
The Normal Curve
This is the standard normal curve, where the average is 0 and the SD is 1
The Normal Curve
Though the equation used to draw the curve is not easy to work with, there is a table of values for the standard normal distribution We will use this table to find areas under
the curve The table is on page A-105 of your text
The Normal Curve
Properties of the standard normal curve The curve is “bell-shaped” with its highest point
at 0
It is symmetric about a vertical line through 0
The curve approaches the horizontal axis, but the curve and the horizontal axis never meet
The Normal Curve
Area underneath the standard normal curve Half the area lies to the left of 0; half lies to the
right Approximately 68% of the area lies between –1
and 1 Approximately 95% of the area lies between –2
and 2 Approximately 99.7% of the area lies between
–3 and 3
Standardization
Most data sets do not have a mean of 0 and an SD of 1
To be able to use the standard normal curve, we’ll need to standardize numbers in the original data set To standardize a number, subtract the data
set’s average and then divide the difference by the data set’s SD
Standardizing is basically a change of scale Like converting feet to miles
Standardization
Suppose there are two different sections of the same course The scores for the midterm in each section were
approximately normally distributed In first section, the average was 64 and the standard
deviation was 5 Tina scored a 74 in first section
In second section, the average was 72 and the standard deviation was 10 Jack scored an 82 in second section
Which of the two scores is most impressive, relative to the students in his/her section?
Standardization
Convert the following scores in the first section to standard units Alice got a 50
Bob got a 61
Carol got a 64
Dan got a 77
-2.8
-0.6
0
2.6
Standardization
In Jack’s section, students with grades between 62 and 82 received a B What percentage of students in this
section received Bs?
Is this percentage exact?
68.27%
No
Normal Approximation
According to the HANES study, the height of U.S. women was 63.5 inches with an SD of 2.5 inches
Normal Approximation
The normal curve is a smooth-curve histogram for normally distributed data We can estimate percentages within a
given range Find the area under the curve between those
ranges using the standard normal table
Normal Approximation
Sometimes will require cutting and pasting different areas together The standard normal table on page A-
105 takes a standard score z It returns to you the area under the curve
between –z and z
Normal Approximation
Find the area between –1.2 and 1.2 under normal curve
76.99%
Normal Approximation
Find the area between 0 and 1.65 under the standard normal curve
45.055%
Normal Approximation
Find the area between 0 and 3.3 under the standard normal curve
49.9515%
Normal Approximation
Find the area between –0.35 and 0.95 under normal curve
46.58%
Normal Approximation
Find the area between 1.2 and 1.85 under the normal curve
8.29%
Normal Approximation
Find the area between –2.1 and –1.05 under the normal curve
12.9%
Normal Approximation
Find the area to the right of 1 under the normal curve
15.865%
Normal Approximation
Find the area to the left of 0.85 under the normal curve
80.235%
Normal Approximation
If a data set is approximately normal in distribution, we can use the normal curve in place of the data set’s histogram
If you want to estimate the percentage of the data set between two numbers… Standardize the numbers to get z scores Look each z score up in the standard normal
table Cut and paste the areas to match the region you
originally wanted The percentage under the curve will be close to the
percentage in the data set
Normal Approximation
It is generally helpful to sketch the curve first and shade in the desired area This will remind you what the target area
is
Normal Approximation
According to the HANES study, the height of U.S. women was 63.5 inches with an SD of 2.5 inches What percentage of women has heights
between 60 and 68 inches?
88.71%
Normal Approximation
According to the HANES study, the height of U.S. women was 63.5 inches with an SD of 2.5 inches What percentage of women are taller
than 66 inches?
15.865%
Normal Approximation
Sometimes, you will be given the percentage of the data set Want to find score(s) which mark(s) off that
percentage Adjust the area to “center” it
Look up the z score associated with that area in the table
Unstandardize the z score by multiplying it by the SD and adding the average to the product
Normal Approximation
For a certain population of high school students, the SAT-M scores are normally distributed with average 500 and SD=100 A certain engineering college will accept only
high school seniors with SAT-M scores in the top 5%
What is the minimum SAT-M score for this program?
665
Normal Approximation
One way to determine how large a number is in the data set is to find its percentile rank The kth percentile is the value so that k
percent of the data set have values below it
Percentile ranks can be calculated for any data set
Normal Approximation
In one year, the 1600-point SAT scores were approximately normal with an average of 1030 and an SD of 190 If a student scores a 1460, what is her
percentile rank?
98th percentile
Summary
It is often useful to describe a data set with summary statistics The average and median are central tendency statistics
The average is more sensitive to outliers
The standard deviation (SD) is the most common summary statistic for describing a data set’s spread The SD is calculated by taking the RMS of the deviations
from the mean of each data point in the set Most of the points in most data sets will lie within one or
two SD’s from the average
Summary
We can represent a data set graphically by drawing a histogram The percentage of the data set in a bin is the area under
the histogram of that section The height of each block in a histogram is the percentage
of the data in the corresponding bin divided by the width of the bin
The total area under any histogram is 100% The average of a data set is located at the balance
point of the histogram Long tails pull the average in the direction of the tail
Summary
Using the average and SD, we can standardize numbers in the data set The standard score (z) of a number is its
distance from the average in terms of SD’s
We can also take a standard score and convert it back to a raw score
Summary
Many data sets are approximately normal We can estimate the percentage of points in a
data set that fall between two numbers Convert the numbers to standard units Find the area under the standard normal curve by
using the normal table
If a data set is approximately normal, we can use the normal table to estimate percentile ranks